Analytical and Experimental Study of Free Vibration of Beams Carrying Multiple Masses and Springs

Zhuang Wang , Ming Hong*, Junchen Xu and Hongyu Cui

School of Naval Architecture Engineering, Dalian University of Technology, Dalian 116024

1 Introduction1

Some structures in engineering can be simplified into elastic beams carrying multiple concentrated masses and elastic supports with linear springs. It is very significant to design the parameters and arrange the locations of the lumped masses and springs to obtain excellent vibration characteristics of the beams such as the mast on the ship, the outboard frames in the stern and the outfitting pipelines,etc.The designs are mainly related to the effects of the layout of the concentrated stiffness and mass on the vibration system.

There has been extensive research on the vibration analysis of beams carrying concentrated masses or springs at arbitrary locations. Rossitet al.(2001) derived the frequency equation of a cantilever beam with a spring–mass system attached to the free end by the analytical method.Banerjee (2012) solved the same problem by a dynamic stiffness approach. Chang (2000) considered the rotational inertia of the lumped mass, and analyzed the free vibration of a simply supported beam with a constant cross-sectional area carrying a concentrated rigid mass at the mid-point of the beam. Low (2003) derived the frequency equation of an Euler-Bernoulli type beam carrying multiple concentrated masses at an arbitrary location by using two methods:frequency determinant and Rayleigh quotient. The results of using the two methods were compared and discussed as well.Then Low (2001) used the Laplace transform method to solve the same problem again. Xiaet al. (1999) deduced the analytical expressions of the lateral vibration characteristics of a beam with arbitrary lumped masses and elastic spring supports by using the Laplace transform. However, the expressions were too complex so that they were only applicable to the calculation of the beam carrying few masses and springs. Based on the work of Low’s, a new analytical expression of the frequency equation and the modal function of the lateral vibration of a beam with lumped masses was derived by means of symbol operation,but the case of springs was neglected (Penget al., 2002).Wuet al.(1998) determined the natural frequencies and the corresponding mode shapes of a uniform cantilever beam carrying any number of elastically mounted point masses by means of the analytical and numerical combined method(ANCM). Liet al.(2012) obtained the natural frequencies and the modal mode of a flexible beam attaching multiple absorbers by use of the Ritz method. All these research efforts were focused on solving the eigenvalue problems of the beams without considering the influence of concentrated parameters such as the locations of the lumped elements, the stiffness of the springs and the weight of masses. But these studies represent great importance to help with the guidance of the design and to avoid resonance in engineering.

Based on the Laplace transform method, this paper derives the mode shape functions and the frequency equations of the beams carrying multiple lumped masses and elastic spring supports in the typical boundary conditions. A cantilever beam with one concentrated mass and one spring is selected to obtain its natural frequencies and corresponding mode shapes. In order to verify the correctness of the analytical results, an experiment is carried out to get the modal parameters of the beam based on the NExT-ERA method (Jianget al., 2011). By comparing the analytical and experimental results, the influence of the concentrated parameters like the locations of the lumped elements, the stiffness of the springs and the weight of masses on natural frequencies are discussed. This data provides a reference for the vibration analysis methods and lumped parameters layout design of the elastic beams in engineering.

2 Mode shape functions and the frequency equations of the beam

Concerning the uniform elastic beam, the mass and stiffness distribution in the structure is uniformly continuous.The vibration differential equation can be determined through the stress analysis of a micro-section of the beam.And the natural frequencies and mode shape can be obtained by the method of separation of variables. If the beam carries any concentrated masses and concentrated stiffness, its mass and stiffness distribution will no longer be continuous.Therefore, the Dirac delta function (δ) can be introduced to represent the effect of the concentrated masses and stiffness along the beam with various boundary conditions. Due to the existence of the Dirac delta function (δ), the solution of the differential equation becomes complicated.

2.1 Mode shape functions

Fig.1 Elastic beam with lumped masses and springs

Referring to Fig.1, consider it as a uniform elastic beam with a length ofLcarryingrconcentrated masses andslinear springs. WhereMiandKjare the spatial coordinates of theith mass andjth spring along the beam. The differential equation for the lateral vibration of the beam can be given in the following form:

whereEis the Young’s modulus,Ithe moment of inertia of the beam’s cross-sectional area,Aρthe mass per unit length of the beam,w(x,t) the transverse deflection of the beam at positionand timet,Mithe mass of theith lumped mass,Kjthe stiffness of thejth spring, andδ(?) the Dirac delta function.Eq.

The transverse deflection of the beam may be assumed to bew(x,t)=Y(x) sinωt, wherex=x L, so the differential (1) can be written as:

By taking the Laplace transform of Eq. (2), we can have:

Then taking the inverse Laplace transform of Eq.(3), one obtains:

whereH(x?xMi)andH(x?xKj)are the unit step functions atx=xMiandx=xKj, respectively.

For convenience, we use the symbol substitution in the forms:

Then they satisfy the following relations:

To substitute Eq.(5) with Eq.(4), one obtains the mode shape function:

2.2 The frequency equation of the beam with a mass and a spring in the middle

To derive the frequency equation, let us assume there are only one concentrated mass and a spring in the middle of the beam, hence in Eq.(1), we haver= 1,s= 1,xM1=1 2 andxK1=1 2.

Consider the clamped-free boundary condition as defined byY(0)=Y′(0) = 0，Y′(1)=Y′(1) = 0，and substitutexM1andxK1into the derived functions of Eq.(7), respectively,one can obtain:

The Eq.(10) and Eq.(11) can be written in matrix form:

where

From the determinant of the matrix in Eq.(12) we can get the transcendental expression of the frequency equation:

If the beam is simply supported, the frequency equation can be derived with the following form:

We can get the frequency equations when the beam is in other boundary conditions by the same approach. It is unnecessary to go into details any more.

2.3 The frequency equation of the beam with a mass and a spring at an arbitrary location

In fact, the concentrated mass and spring in the structure can be in any position. So it is necessary to know the variation of the modal parameters of the beam when the masses and springs are in different positions. Consider a cantilever beam, where1Lξandξ2Lare the coordinates of the mass and spring inxdirection, respectively, as shown in Fig.2.

Fig.2 Beam with lumped masses and lumped springs in random position

From Eq.(7) and the boundary condition of the cantilever beam, we have the mode shape function:

Whenξ1>gt;ξ2, according to the derivation in the previous section one can obtain the determinant of the frequency equation:

where,

where,

3 Modal parameters identification by the natural excitation technique

The natural excitation technique-eigensystem realization algorithm (NExT-ERA) is taken to get the modal parameters of the beam in the experiment in this paper.

3.1 Natural excitation technique(NExT)

NExT is a technique used to identify modal parameters under ambient excitation in time domain. This method is based on the assumption that when the system is excited by stationary white noise, the auto- or cross-correlation function of response signals and the impulse response function of the structure have a similar expression.Therefore, the impulse response functions can be replaced by the auto- or cross-correlation function of response signals(Jameset al., 1995).

3.2 Eigensystem realization algorithm (ERA)

The ERA method utilizes the impulse response signals to build the generalized Hankel matrix. The minimal realization of the system is obtained by the singular value decomposition of the Hankel matrix. Based on the minimal order system matrix, one can identify the modal parameters of the system (Xuet al., 2012).

The state space expression of a linear and time discrete system at instant timetk+1is given by:

wherekrepresents the discrete time,x(k)is the state vector,u(k) =u(kΔt) andy(k) =y(kΔt) are input and output vectors, respectively.A,BandCare the system, input and output matrices, respectively.

Assuming that the number of impulse inputs on the system isS, and the number of output sensors isM, the mathematical model of ERA can be described as:

whereh(k) is the impulse response matrix. Forming the Hankel matrix:

whereαandβare arbitrary integers. Whenk=1 , one obtains the singular value decomposition (SVD) of Hankel matrixH(0):

The eigenvalue decomposition (EVD) of the discrete time system matrixAcan be written as:

whereZandψare the eigenvalue matrix and eigenvector matrix ofA, respectively.

The eigenvectors of the continuous-time system matrixC Aare the same as those ofA. And the eigenvalue of them satisfies the relationship:

whereziis the diagonal element of the eigenvalue matrixZ, andiλthe eigenvalue ofcA.

According to the modal theory, the modal frequency, the modal damping ratio and the modal shape matrix are in the following forms:

4 Comparisons of numerical and experimental results and discussions

4.1 Effects of the mass coefficient and stiffness coefficient on natural frequencies

In order to study the influences of the size of mass and the stiffness of the spring on natural frequency, using the above theory, numerical results are obtained from a cantilever beam with a lumped mass at the tip and a spring in the middle, as shown in Fig.4. The mass coefficientαand stiffness coefficientβhave three different values,respectively. To make a comparison between the analytical and experimental results, the material parameters of the beam are all given the same values as that in the experiment,as shown in Table 1 and Table 2. The first four natural frequenciesanωcomputed analytically for different values ofαandβare shown in Tables 3 - 5.An experiment is carried out to identify the modal parameters of the cantilever beam based on the above NExT-ERA method as shown in Fig.3 and Fig.4. The material properties of the beam are shown in Table 1. The three different values of mass and stiffness used are shown in Table 2. In the experiment, the mass and the spring can be fixed in the beam at arbitrary positions. A white noise excitation is applied to the beam near the clamped end. The response of acceleration can be obtained by the sensors. The first four natural frequenciesexωidentified in the experiment for different values ofαandβare also listed in Tables 3 - 5.

In addition, in order to verify the correctness of the analytical and experimental results further, the frequenciesωFEMobtained by ANSYS are also shown in the Tables.

Fig.3 Testing ground

Fig.4 The experimental model

Table 1 Material properties of the beam

Table 2 Properties of the masses and springs

From Table 3, Table 4 and Table 5, it can be seen that the analytical results agree well with the experimental ones. And the numerical ones obtained by ANSYS further prove the correctness of the analytical approach in this paper.

Referring to the results shown in the Tabs, it is clear that when the positions of the mass and spring are fixed and the value ofαis kept constant, the natural frequencies increase as the value ofβincreases. If the value ofβis kept constant, the natural frequencies decrease as the value ofαincreases. As we know, according to the single degree of freedom system (SDOF), its natural frequency is proportional to the stiffness of the system, and inversely proportional to the mass. And the variations of the natural frequencies of the beam also comply with this rule.

It should be noted that the experimental results of the natural frequencies will be a little less than the analytical ones due to the effect of the mass of the sensors.

Table 3 The first four natural frequencies when α=0.07568,β has different values Hz

Table 4 The first four natural frequencies when α=0.12339,β has different values Hz

Table 5 The first four natural frequencies when α=0.18509,β has different values Hz

From Eq.(15), with the condition ofr=1,s=1,xM1=1 andxK1=1 2, the mode shape function can be written as:

where theY′(0) andY′(0) can be obtained from Eq.(12).

Whenα=0.12339 andβ=21.7612, one can have the first four mode shapes as shown in Fig.5. And the identified ones by experiment are shown in Fig.6. It is clear that the results agree well with each other.

Fig.5 The analytical first four mode shapes

Fig.6 The first four mode shapes obtained from experiment

4.2 Effects of the locations of the mass and spring on natural frequencies

To study the effects of the locations of the spring on natural frequencies, we have the beam with a mass fixed in the end and the spring in nine different positions along the beam. This means that we haveξ1=1 andξ2= 0.1, 0 .2, . ... ,0 .9 in the Eq.(16) and Eq.(17). The mass and spring coefficient areα1=0.12339andβ1=21.7612,respectively. The first four natural frequencies for different values ofβare computed analytically. Moreover, the results are also obtained by experiment so as to be compared with the analytical ones.

For presentational purposes, the curves are plotted which indicates the variation of each order of natural frequencyiωwith the spring moving from one end to the other as shown in Fig.7.

Similarly, whenξ2=1 andξ1=0.1, 0 .2, . ... ,0 .9 in the Eq.(16) and Eq.(17), the changing of each order natural frequencyiωwith the mass moving from the clamped end to another are also obtained as shown in Fig.8.

Referring to the curves illustrated in Fig.7 and Fig.8, it is clear that the analytical results agree well with the experimental ones. And it has been found that when the spring moves from the clamped end to the free end along the beam, the variation trends ofiωin both sets of results are particularly regular. When comparing the curves with the mode shapes in Fig.5 or Fig.6, it is seen that theω1?ξ2curve is similar to the first order mode shape and theω2?ξ2curve is similar to the second order mode shape.Furthermore, theω3?ξ2curve andω4?ξ2curve comply with the same laws.

Fig.7 The variation of the first four natural frequencies with the spring in different positions

Fig.8 The variation of the first four natural frequencieswith the mass in different positions

These change laws ofiωare not coincidental but predictable. To the first order mode shape of the cantilever beam, its nodal displacement is gradually increasing from the fixed end to the free end. As we know, to a multiple degree of freedom system(MDOF), theith natural frequency is, whereKiis theith modal stiffness,Miis theith modal mass. When the spring moves from the fixed end to the free end, theKiof the beam is changing. Take the first mode of the beam shown in Fig.9(a) and Fig.9(b) as an example. In the condition of (b), the spring located at the position having a larger nodal displacement,leads to the spring having a stronger constraint to the beam than that of in the condition of (a). So the first modal stiffnessK1aof the beam in condition (a) is smaller than the ones atK1bin condition (b). According to the equation above, the first order natural frequency1ωin condition (a) is smaller than the ones in condition (b). The first order modal stiffness 1Kof the beam increases gradually when the spring moves from the fixed end to the free end. So the first order natural frequency1ωis increasing as the2ξis changing. For the second order mode shape, the peak displacement is in the middle of the beam, so the constraint on the beam first increases and then decreases which leads to the result that the second order modal stiffnessK2increases first and then decreases. As a result, the second order natural frequency2ωincreases firstly and then decreases, so these are the variations of the third and fourth order natural frequencies.

Fig.9 The modal stiffness when the spring is in different positions

Fig.10 The modal mass when the mass is in different positions

However, when the mass moves along the beam, the changing trends of the frequencies are just the opposite to the ones moving the spring. As shown in Fig.10(a) and Fig.10(b), the first order modal massM1aof the beam in condition (a) is smaller thanM1bin condition (b), so the first natural frequency1ωin condition (a) is bigger than the ones in condition (b). With an analogy of this, the mass moving from the fixed end to the free end increases the first order modal massM1, which decreases the first order natural frequency1ω. For the second mode, the second order modal massM2increases first and then decreases. So the second order natural frequency2ωfirst decreases and then increases.

5 Conclusion

The structures in engineering can be simplified into elastic beams with concentrated masses and lumped elastic supports. Studying the law of vibration of these beams can help guide their design and avoid resonance. In this paper,an analytical approach and a modal identification experiment were taken at the same time to obtain the mode shape functions and frequency equations of the beams carrying multiple concentrated masses and springs. And the modal analysis experiment was carried out by use of the NExT-ERA method to identify the modal parameters of the beam.

The variation of the natural frequencies of the beam is discussed by comparing the analytical results with the experimental results. The two sets of results are consistent with each other. This proves the correctness of the analytical approach and the validity of the NExT-ERA method.Concerning the cantilever beam with one mass and a single spring, the effects of the mass coefficient and stiffness coefficient on the natural frequencies is similar to those of the single degree of freedom system. The effects of the locations of the mass and springs on the natural frequencies are related to the orders of the mode shapes.

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